Atlas › Modular forms › Weight 2 › Level 865 › 865.2.a › 865.2.a.c

Newform orbit 865.2.a.c

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$16q - 6q^{2} - 8q^{3} + 18q^{4} - 16q^{5} - 5q^{6} - 12q^{7} - 18q^{8} + 16q^{9} + 6q^{10} - 4q^{11} - 14q^{12} - 9q^{13} + 3q^{14} + 8q^{15} + 18q^{16} - 20q^{17} - 12q^{18} + 5q^{19} - 18q^{20} - 8q^{21} - 6q^{22} - 32q^{23} - 15q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} + 14x^{15} + 46x^{14} - 171x^{13} - 1152x^{12} - 62x^{11} + 9051x^{10} + 8418x^{9} - 32267x^{8} - 40780x^{7} + 55758x^{6} + 75295x^{5} - 47006x^{4} - 58468x^{3} + 17197x^{2} + 15515x - 1457$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
173	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = +1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = -1$ : each L-function in the orbit has odd analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	−6	−8	−16	−12	−4	−9	−20	5	−32	−6	−15	−17	−6	−16	−63	−20	−42	23	−19	−28	−27	2	−36	−39	−14

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.249 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 5, 173, 865]: [0, 1, 15, 88] (certified at ell = 2147482093)
T = [(2, 1), (3, 1)]; h of degree 57 = 1 x 57 (phi(ord eps) x dim S_k^new); 6 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(865, 2)[2]   # 865.2.a.c

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 865:2 1000